#5956464

order	512
nilpotency class	2
isoclinism	isoclinic class of #5955303
automorphism	Aut(G) order 2097152
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2146 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 36 points:

(2,4)(5,6)(7,11)(8,12)(9,10)(13,14,18,17)(15,20,19,16)(21,22)(23,29)(24,32)(25,28)(26,31)(27,33)(30,34)(35,36);(6,10)(11,12)(13,15)(14,16)(17,20)(18,19)(21,22,25,28)(23,29,31,26)(24,27,33,32)(30,35,36,34);(1,2,3,4)(5,6)(7,11)(8,12)(9,10)(13,16,18,20)(14,19,17,15)(21,22)(23,26)(24,27)(25,28)(29,31)(30,34)(32,33)(35,36);(5,7,9,8)(6,11,10,12)(13,14,18,17)(15,16,19,20)(21,23)(22,26)(24,30)(25,31)(27,34)(28,29)(32,35)(33,36);(5,8,9,7)(6,12,10,11)(13,17,18,14)(15,20,19,16)(21,24,25,33)(22,27,28,32)(23,30,31,36)(26,34,29,35)

Polycyclic presentation:

2^9|a^2=fh,b^2=i,c^2=g,d^2=fi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ei,c^b=ch,d^b=di

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 5863660);